Determining the Cutoff Frequency of Two OnePole Filters in Cascade

(a) Often one needs a simple cascaded low-pass \(R C\) filter that will achieve \(-40 \mathrm{~dB} / \mathrm{dec}\). Cascading two identical filters separated by a follower would seem to provide the answer. However, each filter will deliver \(-20 \mathrm{~dB} / \mathrm{dec}\) at the designed frequency, or \(-6 \mathrm{~dB}\). The cutoff frequency will not be that of the individual filter but at a lower frequency. Derive an equation that will allow one to design such a filter so that the cutoff is at the desired frequency.

(b) Test your new equation by designing a cascaded low-pass \(R C\) filter with a desired cutoff frequency of \(1000 \mathrm{rad} / \mathrm{s}\).